Ramsey, Paper, Scissors

We introduce a graph Ramsey game called Ramsey, Paper, Scissors. This game has two players, Proposer and Decider. Starting from an empty graph on n vertices, on each turn Proposer proposes a potential edge and Decider simultaneously decides (without knowing Proposer's choice) whether to add it to the graph. Proposer cannot propose an edge which would create a triangle in the graph. The game ends when Proposer has no legal moves remaining, and Proposer wins if the final graph has independence number at least s . We prove a threshold phenomenon exists for this game by exhibiting randomized strategies for both players that are optimal up to constants. Namely, there exist constants 0 <  A  <  B such that (under optimal play) Proposer wins with high probability if s < A n log n , while Decider wins with high probability if s > B n log n . This is a factor of Θ ( log n ) ) larger than the lower bound coming from the off‐diagonal Ramsey number r (3, s ).